Statistical Quasi Cauchyness on Asymmetric Spaces

Abstract

We call a sequence (xm) of points in an asymmetric metric space (X, d) statistically forward quasi 1 Cauchy if lim (Formula present) for each positive ε, where |A| indicates the cardinality of the set A. We prove that a subset E of X is forward totally bounded if and only if any sequence of points in E has a statistically forward quasi Cauchy subsequence. We also introduce and investigate statistically upward continuity in the sense that a function defined on X into Y is called statistically upward continuous if it preserves statistically forward quasi Cauchy sequences, i.e. (f (xm)) is statistically forward quasi Cauchy whenever (xm) is.